I have this weird theory that he does know who she is because ...well my hypothesis could be wrong and I don't want to give spoilers in case I'm right, so I'll just leave it at that ⁄(⁄ ⁄·⁄ω⁄·⁄ ⁄)⁄
GIVE ITTTTT
Alright, so my theory is that the original duke was involved in some sort of scandal, had to leave town, or got mixed up with a bad situation and had to leave town, marking the disappearance (or death, or some, weird alchemy that turned everyone into trees or something) of the original duke.
Enter the current bishounen duke who comes in and masquerades and takes the vacant position as the Duke and acts the part in full to achieve something.
Current Duke in the past may have been poor, possibly in a bad position, and like earlier in the story where mc girl saves the young boy, she saved him from something. He could have been a product of alchemy, a test subject, which is why they bring the subject up on numerous occasions.
That's why in the garden, he is overcome wth his gratitude/affection (like mc is with her rescuer, the long blonde haired guy she has a crush on), and that's why he says "why did you cut your hair?" referring to the long hair she had when she saved him. He may have also thought he mistook his savior for the man and got them mixed up.
But I also want to point out that when the Duke met the mc on the bridge there was a bright full moon, bright enough that I feel that he would not have confused her for another woman (but also he may not pay attention to the looks but just that they are woman so that may be a possibility, wrong place/wrong time scenario).
So he hires her on as a man and has to keep up treating her like a man for the same reason she keeps that she is a woman, or knows who she works for and wants to keep the opposition close; and so that neither of their secrets are revealed (and so he could poke fun at someone lol).
I may be wrong about him knowing her previously, but I def smell something of a ruse when it comes to the Duke. Alchemy is mentioned enough I def think it is involved. Hope this makes sense--my theory is little far fetched but it's a tentative theory at the moment for me. Sometimes I'm on the mark with my guesses, but sometimes I'm totally off the mark, lol!
I think everyone here is just hoping the mc girl takes enough voice-changing potion so she becomes a man for real and then the comic will become a yaoi (lol)!
I thought he had an idea that she was a girl from the beginning which is why he chose her. Apparently he doesnt and he just flirts with Lindsayan even if he thinks shes a boy.